Research in Scientific Computing in Undergraduate Education

Convergence of a sequence

Given a real number $x$, the sequence $e(n,x)$ is defined recursively by:

$e(1,x)=x$,

$e(2,x)=1$,

$e(n,x)=e(n-1,x)+e(n-2,x)/(n-2)\textrm{ for }n \geq 3.$

(i) Does $\frac{n}{e(n,x)}$ approach a limit as $n$ increases without bound?

(ii) How does this limit, when it exists, depend on $x$?

(iii) When there is a limit, how rapidly does $\frac{n}{e(n,x)}$ converge to the limit? Does this depend on $x$?

This is an investigation in analysis, which is quite hard without recourse to computational methods. It is a nice example of how computational techniques can provide information about a strange sequence.

More generally, what is the behavior of sequences $e(n,x)$ defined recursively by:

$e(1,x)=x$,

$e(2,x)=1$,

$e(n,x)=e(n-1,x)/(an-p)+e(n-2,x)/(bn-q)\textrm{ for }n \geq 3$?